No, this is not remotely true. They'll downvote you into obvlivion on 'Fox News' comments - but this is not cancel culture.
Nobody is 'losing their job' and being 'publicly shamed' because they said the 'like AOC'.
Sure - you might get a few Trumpers in your face for supporting one of the more perceptibly radical faces of the Demcorcatic party - but that's just 'angry culture' - not cancel culture.
" Parler censored people for supporting BLM."
I strongly doubt this is true - even if it is - again, this is not 'cancel culture' - this is just bad groupthink on a site.
If executives and staff were being fired and losing their jobs for showing up at a BLM rally, or donating money to BLM - that would be 'cancel culture' - but I don't believe that is happening.
Mention the name AOC favorably, and you will be shut down. Parler censored people for supporting BLM. Many took pride in tanking Captain Marvel because of its lead's support for feminist causes. Any progressive action you take will cause you to be dubbed "virtue signalling".
It's all just cancel culture. They simply won't call it that.
There's a saying that for the right wing, every accusation is a confession. They're calling it out on the left (much of it nut-picking[1]) because it's a pervasive part of their own culture.
[1] https://rationalwiki.org/wiki/Nutpicking